Monday, July 1, 2013

Thurston's philosophy of proof

I have posted about Henri Poincaré being way ahead of Einstein on relativity. He was also a leader in other areas of mathematical physics. But he was known more as a mathematician, as this essay describes one of his more famous papers:

Algebraic and differential topology have had several episodes of excessively theoretical work. In his history [D], Dieudonné dates the beginning of the field to Poincaré’s Analysis Situs in 1895. This “fascinating and exasperating paper” was extremely intuitive. In spite of its obvious importance it took fifteen or twenty years for real development to begin. Dieudonné expresses surprise at this slow start [D, p.36], but it seems an almost inevitable corollary of how it began: Poincaré claimed too much, proved too little, and his “reckless” methods could not be imitated. The result was a dead area which had to be sorted out before it could take off.

That essay stirred up trouble with this dig at a famous mathematician:

William Thurston’s “geometrization theorem” concerning structures on Haken three-manifolds is another often-cited example. A grand insight delivered with beautiful but insufficient hints, the proof was never fully published. For many investigators this unredeemed claim became a roadblock rather than an inspiration.

About two or three years later, I proved the geometrization theorem for Haken manifolds. It was a hard theorem, and I spent a tremendous amount of effort thinking about it. When I completed the proof, I spent a lot more effort checking the proof, searching for difficulties and testing it against independent information. I’d like to spell out more what I mean when I say I proved this theorem. It meant that I had a clear and complete flow of ideas, including details, that withstood a great deal of scrutiny by myself and by others. Mathematicians have many different styles of thought. My style is not one of making broad sweeping but careless generalities, which are merely hints or inspirations: I make clear mental models, and I think things through. My proofs have turned out to be quite reliable. I have not had trouble backing up claims or producing details for things I have proven. I am good in detecting flaws in my own reasoning as well as in the reasoning of others. However, there is sometimes a huge expansion factor in translating from the encoding in my own thinking to something that can be conveyed to someone else. ...

What mathematicians most wanted and needed from me was to learn my ways of thinking, and not in fact to learn my proof of the geometrization conjecture for Haken manifolds. It is unlikely that the proof of the general geometrization conjecture will consist of pushing the same proof further.

I am not really sure what he is rebutting here, as Thurston seems to concede that he never published the proof. Nevertheless, Thurston's paper is famous, and UCLA mathematician Terry Tao just wrote that he would tag it as a "must read" for all research mathematicians.

Go ahead and read it, but keep in mind that Thurston was on the fringe of mathematics by denying the importance of a written proof. He was cited in a notorious 1993 SciAm article on the Death of Proof, and the author says he relied heavily on Thurston. Thurston was probably embarrassed by this paragraph:

Thurston emphasizes that he believes mathematical truths are discovered and not invented. But on the subject of proofs, he sounds less like a disciple of Plato than of Thomas S. Kuhn, the philosopher who argued in his 1962 book, The Structure of Scientific Revolutions, that scientific theories are accepted for social reasons rather than because they are in any objective sense “true.” “That mathematics reduces in principle to formal proofs is a shaky idea” peculiar to this century, Thurston asserts. “In practice, mathematicians prove theorems in a social context,” he says. “It is a socially conditioned body of knowledge and techniques.”

No mathematician wants to be associated with a Kuhnian paradigm shifter, so this paragraph surely also prompted Thurston's essay. Meanwhile, Thurston's geometrization conjecture has been proved in connection with the solution to the Poincaré conjecture. Yes, Poincare's analysis situs led to a century of research by the smartest people to solve it. And of course the mathematical community has published the proof several times over, even if some of the original provers left some gaps.